Strain wave gearing with full separation of two stresses

ABSTRACT

In a strain wave gearing, the addendum tooth profile of an internal gear is defined by the formula a and that of an external gear is by the formula b at a principal cross-section located at a tooth-trace-direction center of the external gear, on the basis of a movement locus (Mc) of κ=1 of the teeth of the external gear with respect to those of the internal gear in the principle cross-section taken at the center of the tooth trace of the external gear obtained when the tooth meshing is approximated by rack meshing. It is possible to avoid superimposition of flexion-induced bending stresses and tensile stresses caused by load torque at the major-axis locations of the external gear, and the transmission torque capacity of a strain wave gearing can be improved.

TECHNICAL FIELD

The present invention relates to a strain wave gearing, in which aflexible external gear is made to flex into an ellipsoidal shape by awave generator and to partially mesh with a rigid external gear. Morespecifically, the invention relates to a strain wave gearing forimproving the transmission torque capacity thereof by avoidingsuperimposition of bending stress and tensile stress in sections ateither end of the major axis of the ellipsoidal shape of the externalgear, in which the bending stress is produced due to flexion and thetensile stress is caused by load torque due to meshing with the internalgear.

BACKGROUND ART

A strain wave gearing typically has a rigid internal gear, a flexibleexternal gear coaxially arranged inside the internal gear, and a wavegenerator fitted into the external gear. A flat-type strain wave gearingis provided with an external gear having external teeth formed on anouter peripheral surface of a flexible cylinder. The external gear of acup-type or top-hat-type strain wave gearing is provided with a flexiblecylindrical body part, a diaphragm extending in a radial direction froma back end of the cylindrical body part, and external teeth formed in anouter peripheral surface section at the front-end opening side of thecylindrical body part. In a typical strain wave gearing, a circularexternal gear is ellipsoidally flexed by a wave generator, and sectionsat either end of the major axis of the ellipsoidally-flexed externalgear mesh with an internal gear.

Since the invention of the strain wave gearing by its creator, C. W.Musser (Patent document 1), up to the present day, various inventionsbased on the present device have been contrived by numerous researchers,including Mr. Musser, and the present inventor. Even limiting the scopeto inventions relating to tooth profile, various inventions devised. Thepresent inventor, in Patent document 2, proposed to adopt an involutetooth profile as the basic tooth profile, and in Patent documents 3 and4 proposed a tooth profile design method for deriving addendum toothprofiles of an internal gear and an external gear for wide-area contact,using a process of approximating meshing of the two gears by rackmeshing.

In strain wave gearings, a flexible external gear is flexed from a truecircular state to an ellipsoidal shape by a wave generator, andtherefore bending stress due to flexing is produced in sections ateither end of the major axis of the ellipsoidal shape. Onceellipsoidally flexed, an external gear will mesh with the internal gearin these sections at either end of the major axis, thereby giving riseto tensile stress caused by load torque transmitted via the meshingsections. For this reason, high stress is applied on the sections ateither end of the major axis of the external gear (root rim sections),due to the two stresses being superimposed. As is particularly so with alow-gear-ratio strain wave gearing in which both gears have a smallnumber of teeth, there is appreciable flexion of the external gear atthe locations on the major axis; therefore, strong bending stress isproduced in association with the ellipsoidal deformation. Therefore, inorder to improve the transmission torque capacity of a strain wavegearing, it is necessary to reduce the stresses produced in the sectionsat either end of the major axis of the external gear.

In order for stresses produced in sections at either end of the majoraxis of the external gear to be reduced in the prior art, the maximumradial flexion (radial flexion at the major axis location) when theexternal gear is ellipsoidally deformed was set to a flexing amount κmn(κ<1) less than a standard normal flexing amount mn. Here, n is apositive integer, 2n is the difference in the number of teeth betweenthe gears, m is the module of the two gears, and κ is a coefficientknown as the coefficient of deflection (or the coefficient of flexion).Cases in which κ=1 (normal flexion) and the flexing amount is mn, arecalled “non-deflection flexing.” Cases in which the flexing amount κmn(κ<1) in the radial direction is less than mn are called “negativedeflection flexing.” And cases in which the flexing amount κmn (κ>1) inthe radial direction is greater than mn are called “positive deflectionflexing.”

By having the external gear set to negative deflection flexing, bendingstress that occurs in the sections at either end of the major axis ofthe external gear in association with ellipsoidal deformation isreduced. Additionally, by setting the external gear to negativedeflection flexing, the center of meshing of the external gear withrespect to the internal gear is shifted from the sections at either endof the major axis, thereby reducing tensile stress caused by load torqueproduced in the sections at either end of the major axis of the externalgear. Thus, by setting the flexing amount to negative deflection,bending stress caused by flexion in the sections at either end of themajor axis of the external gear is reduced, and superimposition ofbending stress and tensile stress is avoided. A strain wave gearing setto negative deflection flexing was proposed by the present inventor inPatent documents 5 and 6, for example.

PRIOR ART DOCUMENT Patent Document

Patent document 1: U.S. Pat. No. 2,906,143

Patent document 2: JP 45-41171 B

Patent document 3: JP 63-115943 A

Patent document 4: JP 64-79448 A

Patent document 5: JP 4650954 B

Patent document 6: JP 4165810 B

SUMMARY OF THE INVENTION Problems to be Solved by the Invention

In a strain wave gearing, the tooth depth of the two gears relates tothe flexing amount, and where radial flexing amount is set to benegative deflection flexing (κmn, κ<1), which is less than the normaldeflection flexing (=mn, κ=1), the tooth depth becomes smaller. Thesmaller tooth depth poses a risk that ratcheting (tooth jumping) willoccur during high load torque. In order to prevent ratcheting, it isnecessary to make the tooth depth of the two gears as large as possible.

From this standpoint, it would be desirable to be able to separate thesuperimposition of bending stress and tensile stress produced in thesections at either end of the major axis of the external gear whenellipsoidally deformed, while maintaining normal deflection flexingwithout making the flexing amount smaller. However, no rigorousexamination has been made into ways to actively separate superimpositionof bending stress and tensile stress produced in sections at either endof the major axis of the external gear.

An object of the present invention is to provide a strain wave gearing,by which superimposition of bending stress and tensile stress producedin sections at either end of the major axis of an external gear can beavoided substantially and completely, without making the flexing amountof the external gear (the average flexing amount of flexing amount ateach location in the tooth trace direction) smaller than the normalflexing amount. Another object of the present invention is to provide astrain wave gearing, which is able to further increase transmissiontorque capacity thereof by realizing an approximately continuous meshingbetween an external gear and an internal gear along the tooth tracedirection thereof as well as by substantially and completely avoidingsuperimposition of bending stress and tensile stress produced insections at either end of the major axis of the external gear.

Means of Solving the Problems

In order to solve the above problems, in the present invention, the twostresses (bending stress and tensile stress) are separated in asubstantially complete manner by making necessary corrections to thetooth profile of the flexible external gear of a strain wave gearing.

Specifically, the strain wave gearing of the present invention isprovided with a rigid internal gear, a flexible external gear arrangedcoaxially inside the rigid internal gear, and a wave generator fittedinside the flexible external gear,

wherein the external gear is ellipsoidally flexed by the wave generator,and external teeth of the ellipsoidally flexed external gear are causedto mesh with internal teeth of the internal gear in regions avoidingsections at either end in the major axis direction of the ellipsoidallyflexed external gear;

The internal gear, and the external gear prior to ellipsoidaldeformation, both are spur gears of module m;

the number of teeth of the external gear is fewer by 2n than the numberof teeth of the internal gear, where n is a positive integer;

at a major axis location on an ellipsoidal rim neutral curve of theexternal gear in an axis-perpendicular cross-section at a prescribedlocation lying in the tooth trace direction of the external gear, radialflexing amount with respect to the rim neutral circle prior to flexionis 2κmn, where κ is a deflection coefficient, and where anaxis-perpendicular cross-section established at a prescribed locationlying in the tooth trace direction of the external gear is taken as aprincipal cross-section, the principal cross-section is a non-deflectionflexing cross-section in which the deflection coefficient κ=1;

on the basis of a movement locus of κ=1 by the teeth of the externalgear with respect to the internal gear, where meshing of the externalgear with respect to the internal gear in the principal cross-section isregarded as rack meshing, the tooth profile of the addendum of theinternal gear is specified by the following formula a:x _(Ca1)=0.25 mn(π+θ−sin θ)y _(Ca1)=0.5 mn(−1+cos θ)  (formula a)where 0≤θ≤π;

the tooth profile of the addendum of the external gear is specified bythe following formula b:x _(Fa1)=0.25 mn[π−θ+sin θ−ε{cos(θ/2)−sin(θ/2)}]y _(Fa1)=mn[0.5(1−cos θ)−(ε/4){sin(θ/2)−cos(θ/2)}](formula b)where 0≤ε≤0.1 and 0≤θ≤π; and

the tooth profiles of the dedenda of each of the internal gear and theexternal gear are set to any shape that does not interfere with thetooth profile of the addendum of the other gear.

In the case of a flat-type strain wave gearing, the tooth profile of theaddendums of the internal gear, in axis-perpendicular cross-sectionsthereof along the tooth trace direction, is defined by the above formulaa, and the profile of the addendums of the external gear, inaxis-perpendicular cross-sections thereof along the tooth tracedirection, is defined by the above formula b.

In the case of a cup-type strain wave gearing or a top-hat-type strainwave gearing, the external gear is provided with a flexible cylindricalbody part, and a diaphragm extending in a radial direction from the backend of this cylindrical body part. The external teeth are formed in anouter peripheral section at the front open-end side of the cylindricalbody part. The flexing amount of the external teeth changes inproportion to the distance from the diaphragm from the external-teethinner end at the diaphragm side towards the external-teeth open end atthe front open-end side in the tooth trace direction. The principalcross-section is located at the tooth-trace-direction center between theexternal-teeth open end and the external-teeth inner end of the externalteeth.

In this case, the addendum tooth profile of the internal gear is definedby the aforementioned formula a. Whereas, the addendum tooth profile ofthe external gear in the principal cross-section is defined by theaforementioned formula b. The tooth profile in axis-perpendicularcross-sections, other than the principal cross-section, in the toothtrace direction in the external gear are shifted profiles in which thetooth profile of the principal cross-section is subjected to shiftingaccording to the flexing amount of each of the axis-perpendicularcross-sections. Specifically, the tooth profiles of axis-perpendicularcross-sections of the tooth trace direction, from the principalcross-section to the external-teeth open end of the external gear, areobtained by subjecting the tooth profile of the principal cross-sectionto shifting, in such a way that apex portions of the κ>1 movement locusdescribed by the tooth profile in each of the axis-perpendicularcross-sections contact apex portions of the κ=1 movement locus in theprincipal cross-section. The tooth profiles of axis-perpendicularcross-sections of the tooth trace direction, from the principalcross-section to the external-teeth inner end of the external gear, areobtained by subjecting the tooth profile of the principal cross-sectionto shifting, in such a way that nadir portions of the κ<1 movement locusdescribed by the tooth profiles in the axis-perpendicular cross-sectionscontact nadir portions of the κ=1 movement locus in the principalcross-section.

Effect of the Invention

According to the present invention, it is possible to avoid superimposedflexion-induced bending stresses and tensile stresses caused by loadtorque arising at the major-axis locations on the ellipsoidal rimneutral curve of the external gear on an axis-perpendicularcross-section having a deflection coefficient κ=1 (principalcross-section) in the external gear of the strain wave gearing, wherebysubstantially and completely separating the bending stress and thetensile stresses. Therefore, the transmission torque capacity of astrain wave gearing can be improved, without the need to adopt negativedeflection flexing having a deflection coefficient κ<1 in a flat-typestrain wave gearing, and without the need to adopt negative deflectionflexing having a deflection coefficient κ<1 along the entire toothprofile in a cup-type or top-hat-type strain wave gearing.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a simplified front view showing an example of a strain wavegearing according to the present invention;

FIG. 2 is an illustrative representation of flexion conditions of anexternal gear of cup and top hat profiles, where (a) shows a state priorto deformation, (b) shows a state of a cross-section including the majoraxis of an ellipsoidally deformed external gear, and (c) shows a stateof a cross-section including the minor axis of an ellipsoidally deformedexternal gear;

FIG. 3A is a graph showing movement loci of external gear teeth withrespect to internal gear teeth, obtained in a case in which meshing ofteeth of the external gear with respect to the internal gear isapproximated by rack meshing, in an external-teeth inner end (κ<1), aprincipal cross-section (κ=1), and an external-teeth open end (κ>1) inthe tooth trace direction of an external gear;

FIG. 3B is a graph showing movement loci of external gear teeth withrespect to internal gear teeth, obtained in a case in which meshing ofteeth of the external gear with respect to the internal gear isapproximated by rack meshing, in an external-teeth inner end (κ<1), aprincipal cross-section (κ=1) and an external-teeth open end (κ>1) inthe trace direction of an external gear after being subjected toshifting;

FIG. 4A is an illustrative representation of a tooth profile of anaddendum of an internal gear;

FIG. 4B is an illustrative representation of a tooth profile of anaddendum in a principal cross-section of an external gear;

FIG. 4C is an illustrative representation of an example of a toothprofile of a dedendum of an internal gear;

FIG. 4D is an illustrative representation of an example of a toothprofile of a dedendum of an external gear;

FIG. 4E is an illustrative representation of an example of a toothprofile of a dedendum of an internal gear;

FIG. 5 is an illustrative representation of tooth profiles of anexternal gear and an internal gear in a principal cross-section;

FIG. 6 is a graph showing an example of the amount of shifting near aprincipal cross-section in the tooth trace direction of an externalgear;

FIG. 7 is an illustrative representation of tooth profile contours inthe tooth trace direction of an external gear having undergone shifting;

FIG. 8 is an illustrative representation of meshing of external gearteeth with respect to internal gear teeth in an external-teeth open endof an external gear;

FIG. 9 is (a) an illustrative representation of meshing of external gearteeth with respect to internal gear teeth in a principal cross-sectionof an external gear, and (b) a partial enlarged view thereof; and

FIG. 10 is an illustrative representation of meshing of external gearteeth with respect to internal gear teeth in an external-teeth open endof an external gear.

MODE FOR CARRYING OUT THE INVENTION Configuration of Strain Wave Gearing

FIG. 1 is a front view of a strain wave gearing according to the presentinvention. FIG. 2 (a) to (c) are cross-sectional views showingconditions in which an open-end portion of a flexible external gear inthe strain wave gearing is ellipsoidally flexed; FIG. 2 (a) shows astate prior to deformation, FIG. 2 (b) shows a cross-section includingthe major axis of an ellipsoidal curve subsequent to deformation, andFIG. 2 (c) shows a cross-section including the minor axis of anellipsoidal curve subsequent to deformation, respectively. In FIGS. 2(a) to (c), the solid lines represent the diaphragm and boss sections ofa flexible external gear having a cup profile, and broken linesrepresent the diaphragm and boss sections of a flexible external gearhaving a top hat profile.

As shown in the drawings, the strain wave gearing 1 has an annular,rigid internal gear 2, a flexible external gear 3 arranged to the insideof the internal gear, and a wave generator 4 of ellipsoidal profile,fitted into the inside of the external gear. The internal gear 2 and thepre-deformation external gear 3 are spur gears of module m. Thedifference in the number of teeth between the internal gear 2 and theexternal gear 3 is 2n (n is a positive integer), and the circularexternal gear 3 of the strain wave gearing 1 is ellipsoidally flexed bythe wave generator 4 of ellipsoidal profile. External teeth 34 of theexternal gear 3 (hereinafter, in some instances termed simply “teeth34”) and internal teeth 24 of the internal gear 2 (hereinafter, in someinstances termed simply “teeth 24”) mesh with one another at positionsor regions apart from both end sections in the direction of a major axisLa of the external gear 3 when ellipsoidally flexed.

As the wave generator 4 rotates, the location of meshing by the twogears 2, 3 moves in a circumferential direction, and the two gears 2, 3rotate in relative fashion in accordance with the difference in thenumber of teeth of the gears. The external gear 31 is provided with aflexible cylindrical body part 31, a diaphragm 32 continuous with flarein a radial direction from a back end 31 b which is one end of thecylindrical body part 31, a boss 33 continuous with the diaphragm 32,and the external teeth 34, which are formed in an outer peripheralsurface section at a front-end opening 31 a side at the other end of thecylindrical body part 31.

The ellipsoidal-profile wave generator 4 is fitted into an innerperipheral surface section of the external-teeth formation section ofthe cylindrical body part 31. The wave generator 4 causes thecylindrical body part 31 to undergo a gradual increase in flexiontowards the outside or the inside in the radial direction, towards thefront-end opening 31 a from the back end 31 b at the diaphragm side. Asshown in FIG. 2 (b), in a cross-section that includes the major axis Laof the ellipsoidal curve (see FIG. 1), the flexing amount towards theradial outside gradually increases in a substantially proportionalrelationship to the distance from the back end 31 b to the front-endopening 31 a. As shown in FIG. 2 (c), in a cross-section that includesthe minor axis Lb of the ellipsoidal curve (see FIG. 1), the flexingamount towards the radial inside gradually increases in a substantiallyproportional relationship to the distance from the back end 31 b to thefront-end opening 31 a. The external teeth 34 formed in an outerperipheral surface section at the front-end opening 31 a side likewiseexperience a gradually increasing amount of flexion, substantiallyproportional to the distance from the back end 31 b, going from anexternal-teeth inner end portion 34 b to an external-teeth open endportion 34 a in the tooth trace direction.

In an axis-perpendicular cross-section at any location in the toothtrace direction of the external gear 34, a circle passing through thecenter in the thickness direction of the root rim of the external gear34 prior to ellipsoidal flexing is a rim neutral circle. On the otherhand, an ellipsoidal curve passing through the center in the thicknessdirection of the root rim after ellipsoidal flexing is termed a “rimneutral curve.” The flexing amount w in the major axis direction withrespect to the rim neutral circle at a major axis location on theellipsoidal rim neutral curve is represented by 2κmn, where κ is adeflection coefficient (a real number including 1).

Specifically, where the number of external teeth 34 of the external gear3 is denoted by Z_(F), the number of internal teeth 24 of the internalgear 2 by Z_(C), and the gear ratio of the strain wave gearing 1 by R(=Z_(F)/(Z_(C)−Z_(F))=Z_(F)/2n), the value (mZ_(F)/R=2 mn) obtained bydividing the pitch circle diameter mZ_(F) of the external gear 3 by thegear ratio R is the regular (standard) flexing amount w₀ (=2 mn) in themajor axis direction. The strain wave gearing 1 is typically designed toinduce flexion by the regular amount flexion w₀, in a region where theball center of a wave bearing of the wave generator 4 is located in thetooth trace direction of the external gear 3, and normally at a locationin a center portion in the tooth trace direction of the external gear.

The deflection coefficient κ represents a value obtained by dividing theflexing amount w in axis-perpendicular cross-sections in the tooth widthdirection of the external gear 3, by the regular flexing amount.Consequently, in the external gear 34, the deflection coefficient at thelocation at which the regular flexing amount w₀ is obtained is κ=1, thedeflection coefficient at cross-sectional locations of lesser flexingamounts w is κ<1, and the deflection coefficient at cross-sectionallocations of greater flexing amounts w is κ>1. A tooth profile withwhich the regular flexing amount w₀ (κ=1) is obtained in the externalgear 34 is termed a “non-deflection tooth profile,” a tooth profile withwhich a flexing amount less than the regular flexing amount (κ<1) isobtained is termed a “negative deflection tooth profile,” and a toothprofile with which a flexing amount greater than the regular amountflexion (κ>1) is obtained is termed a “positive deflection toothprofile.” In the present example, an axis-perpendicular cross-section ina center portion in the tooth trace direction of the external gear 34 isestablished as the principal cross-section 34 c in which κ=1.

FIG. 3A is a diagram showing movement loci of the teeth 34 of theexternal gear 3 with respect to the teeth 24 of the internal gear 2,obtained in a case in which relative motion of the gears 2, 3 of thestrain wave gearing 1 is rack approximated. In the drawing, the x axisindicates the direction of translation of the rack, and the y axisindicates a direction perpendicular thereto. The origin of the y axis isthe average position of amplitude of the movement loci. Curve Ma is amovement locus obtained at the external-teeth open end portion 34 a, andcurve Mb is a movement locus obtained at the external-teeth inner endportion 34 b. Curve Mc is a movement locus obtained at any location fromthe external-teeth open end portion 34 a to the external-teeth inner endportion 34 b in the tooth trace direction, and in the present example isobtained in a center portion in the tooth trace direction. Theaxis-perpendicular cross-section at this location is referred tohereinafter as “principal cross-section 34 c.” The movement locus of theteeth 34 of the external gear 3 with respect to the teeth 24 of theinternal gear is given by the following formulas.x _(Fa)=0.5 mn(θ−κ sin θ)y _(Fa)=κmn cos θ

To simplify the description, the above formulas are represented by thefollowing formula (1), where module m=1 and n=1 (difference in number ofteeth 2n=2).x _(Fa)=0.5(θ−κ sin θ)y _(Fa)=κ cos θ  (Formula 1)

Method for Forming Tooth Profile in Principal Cross-Section

A tooth profile of the addendums of the internal teeth 24 in theprincipal cross-section 34 c (deflection coefficient κ=1), afforded byrack approximation, will be described. The movement locus Mc obtained inthe principal cross-section 34 c in the external gear 34 is utilized inorder to specify the addendum profile of the internal teeth 24 in theprincipal cross-section 34 c.

First, in the movement locus Mc in the principal cross-section 34 c ofFIG. 3A, a first curve AB for which the range of parameter θ is π to 0is selected. The location at which the parameter θ=π is point B, whichis the nadir point of the movement locus Mc; point A at which theparameter θ=0 is the apex point of the movement locus Mc. Subsequently,a λ-fold (0<λ<1) similarity transformation of the first curve AB withpoint B as the center of similarity gives a first similarity curve BC(see FIG. 4A). The first similarity curve BC is utilized as the addendumprofile for the teeth 24 of the rigid internal gear 2. In the presentexample, λ is 0.5.

The addendum profile for the teeth 24 of the internal gear 2 establishedin this manner is given by the following formula 2.x _(Ca1)=0.5{(1−λ)π+λ(θ−κ sin θ)}y _(Ca1)=κ{λ(1+cos θ)−1}  (Formula 2)where 0≤θ≤π.

Since λ=0.5 and κ=1, substituting these into formula 2 gives formula 2A.FIG. 4A shows a first similarity curve BC given by formula 2A, the curvebeing an addendum tooth profile curve 24C1 for the internal gear 2.

Internal Gear Addendum Profile

x _(Ca1)=0.25(π+θ−sin θ)y _(Ca1)=0.5(−1+cos θ)  (Formula 2A)where 0≤θ≤π

Subsequently, the first similarity curve BC undergoes 180° rotation and(1−λ)-fold similarity transformation, with point C, which is the endpoint at the opposite side from point B in the first similarity curveBC, as the center, to obtain a second similarity curve. The secondsimilarity curve is given by the following formula 3.x(θ)=0.5{(1−λ)(π−θ+κ sin θ)}y(θ)=κ{(λ−1)(1−cos θ)}  (Formula 3)where 0≤θ≤π

Since λ=0.5 and κ=1, substituting these into formula 2 gives formula 3A.The second similarity curve CA given by the formula 3A is shown bydotted line in FIG. 4A.x(θ)=0.25(π−θ+sin θ)}y(θ)=0.5(1−cos θ)  (Formula 3A)where 0≤θ≤π

External Gear Addendum Profile

Here, the addendum profile of the external gear 34 is specified by thefollowing formula 3B. FIG. 4B shows an addendum profile curve 34C1 givenby formula 3B.x _(Fa1)=0.25[π−θ+sin θ−ε{cos(θ/2)−sin(θ/2)}]y _(Fa1)=0.5(1−cos θ)−(ε/4){sin(θ/2)−cos(θ/2)}  (Formula 3B)where 0≤ε≤0.1 and 0≤θ≤π

In formula 3B, meshing of the external gear 3 with the internal gear 2at the major axis La of the ellipsoidal rim neutral curve is eliminatedby introducing the correction term including ε so that, at the majoraxis La, only bending stress due to ellipsoidal flexion is substantiallypresent. The peak of tensile stress due to transmission torque loadappears at the center position (θ=π/4) between the major axis La and theminor axis Lb, which means that the tensile stress is not substantiallygenerated on the major axis La. Therefore, it is possible tosubstantially avoid superimposition of the bending stress and thetensile stress on the both end sections of the major axis of theexternal gear 3 (namely, regions where these stresses are generated canbe separated substantially and completely.

Example of Internal Gear Dedendum Profile

The dedendum profile of each of the two gears 2, 3 may be any profilethat does not give rise to interference with the addendum profile of thecounterpart gear. For example, the dedendum profile of the internal gear2 can be such that a curve created in the internal gear 2 during theinterval that the addendum profile of the external gear 3 moves from theapex point to the nadir point of the movement locus Mc is defined as thededendum profile of maximum tooth thickness of the internal gear 2. Thisdedendum profile is given by the following formula 4. FIG. 4C shows adedendum profile curve 24C2 given by formula 4.x _(Ca2)=0.25(π−θ+sin θ)y _(Ca2)=0.5(1−cos θ)}  (Formula 4)where 0≤θ≤π

Likewise, the curve that the addendum profile of the internal gear 2creates in the external gear 3 during the interval that the addendumprofile of the external gear 3 moves from the apex point to the nadirpoint of the movement locus Mc can be defined as the dedendum profile ofmaximum tooth thickness of the external gear 3. This dedendum profile isgiven by the following formula 5. FIG. 4D shows a dedendum profile curve34C2 given by formula 5.x _(Fa2)=0.25[π−θ+sin θ−ε{cos(θ/2)−sin(θ/2)}]y _(Fa2)=0.5(−1+cos θ)−(ε/4){sin(θ/2)−cos(θ/2)}  (Formula 5)where 0≤ε≤0.1 and 0≤θ≤π

FIG. 4E shows, in an enlarged manner, the dedendum profile of theinternal gear 2 defined by the profile curve 24C2 and the addendumprofile of the external gear 3 defied by the profile curve 34C1, inwhich profile modification applied to the addendum profile of theexternal gear 3 is depicted.

FIG. 5 shows an external tooth profile 34C and an internal tooth profile24C defined by meshing of the aforementioned individual tooth profilesin the principal cross-sections 34 c of the external gear and theinternal gear.

Tooth Profiles in Axis-Perpendicular Cross-Sections Other than PrincipalCross-Sections

In a flat-type strain wave gearing, the tooth profiles ofaxis-perpendicular cross-sections in the tooth trace direction of theinternal gear 2 and the external gear 3 are the same as the toothprofiles in the principal cross-section 34 c established as describedabove.

By contrast, in a cup-type strain wave gearing or a top-hat-type strainwave gearing, tooth profiles of axis-perpendicular cross-sections in thetooth trace direction of the internal gear 2 are identical to the toothprofile at the location of the principal cross-section 34 c establishedas described above. However, tooth profiles of axis-perpendicularcross-sections other than the principal cross-section 34 c in the toothtrace direction of the external gear 3 are shifted profiles in which thetooth profile of the principal cross-section 34 c has been subjected toshifting according to the flexing amount of each axis-perpendicularcross-section.

Specifically, the tooth profiles of axis-perpendicular cross-sections inthe tooth trace direction from the principal cross-section 34 c to theexternal-teeth open end portion 34 a of the external gear 3 are toothprofiles obtained when the external tooth profile 34C of the principalcross-section 34 c undergoes shifting such that apex portions of κ>1movement loci described by the external teeth 34 in axis-perpendicularcross-sections contact an apex portion of the κ=1 movement locus in theprincipal cross-section 34 c. The tooth profiles of axis-perpendicularcross-sections in the tooth trace direction from the principalcross-section 34 c to the external-teeth inner end portion 34 b of theexternal teeth 34 c are tooth profiles obtained when the external toothprofile 34C of the principal cross-section 34 c undergoes shifting suchthat nadir portions of κ<1 movement loci described by the external teeth34 in axis-perpendicular cross-sections contact a nadir portion of theκ=1 movement locus in the principal cross-section 34 c.

In specific terms, tooth profiles of cross-sections in the tooth tracedirection, other than the principal section, in the external gear 3 areestablished as follows. As shown in FIG. 3B, in axis-perpendicularcross-sections at locations from the principal cross-section 34 c to theexternal-teeth open end portion 34 a, in which the deflectioncoefficient is κ>1, the amount of shifting h of the teeth 34 of theexternal gear 3 is given by the following formulas 6, such that an apexportion of a movement locus Ma1 derived by rack approximation of theteeth 34 of the external gear 3 with respect to the teeth 24 of theinternal gear contacts a movement locus Mc in the principalcross-section 34 c.h=λ(κ)(κ−1)  (Formulas 6)

As noted above, a rack-approximated movement locus of the teeth 34 ofthe external gear 3 with respect to the teeth 24 of the internal gear 2in axis-perpendicular cross-sections of the external gear in which thedeflection coefficient κ is 1 or greater is indicated by the followingformula.x _(Fa)=0.5(θ−κ sin θ)y _(Fa)=κ cos θ  (Formula A)

A pressure angle α_(κ) of a tangent to a movement locus, with respect toa point on the movement locus, is indicated by the following formula.tan α_(κ)=0.5(1−κ cos θ_(κ))/κ sin θ_(κ)  (Formula B)

A pressure angle α₁ of a tangent with respect to a point on the κ=1movement locus is indicated by the following formula.tan α₁=0.5(1−cos θ₁)/sin θ₁  (Formula C)

The pressure angles are thereby equated to obtain the following formula.(1−κ cos θ_(κ))/κ sin θ_(κ)−(1−cos θ₁)/sin θ₁=0  (Formula D)

Next, the x coordinates of the contact points are equated to obtain thefollowing formula.θ_(κ)−κ sin θ_(κ)−θ₁+sin θ₁=0  (Formula E)

Here, by simultaneously solving formula D and formula E, and calculatingθ_(κ) and θ₁, the amount of shifting h is calculated from the followingformula.h=κ cos θ_(κ)−cos θ₁λ(κ)=h/(κ−1)  (Formula F)

Next, in axis-perpendicular cross-sections situated at locations fromthe principal cross-section 34 c to the external-teeth inner end portion34 b of the external gear 3 and in which the deflection coefficient isκ<1, the teeth 34 of the external gear 3 are shifted such that a nadirportion of a movement locus Mb1 of the teeth 34 of the external gear 3with respect to the teeth 24 of the internal gear 2 contacts a nadirportion of the movement locus Mc in the principal cross-section 34 c, asshown in FIG. 3B. The magnitude of shifting at this time is given by thefollowing formula.h=κ−1

FIG. 6 is a graph showing an example of the amount of shifting near theprincipal cross-section in the tooth trace direction of the externalgear 3. The horizontal axis in the drawing indicates the distance fromthe center portion in the tooth trace direction of the external teeth 34(the principal cross-section 34 c), and the vertical axis indicates theamount of shifting h. The amount of shifting h is indicated by straightshifting lines L1, L2 of identical slope. The straight shifting line L1indicates the amount of shifting from the principal cross-section 34 tothe external-teeth open end portion 34 a, and the straight shifting lineL2 indicates the amount of shifting from the principal cross-section 34to the external-teeth inner end portion 34 b.

A quartic curve C1 having the principal cross-section 34 c as the apexpoint and contacting the straight shifting lines L1, L2 is also shown inFIG. 6. When amounts of shifting in axis-perpendicular cross-sectionsare determined on the basis of this quartic curve C1, a substantiallyflat portion is formed in a center portion in the tooth trace directionthat includes the principal cross-section 34 c of the external gear 34.In so doing, smoothly varying shifting is ensured, and dimensionmanagement during cutting of the external gear 3 is facilitated.

FIG. 7 is an illustrative representation of the tooth profile outlinesof the inner gear 24 and the external gear 34, in which the toothprofile outline of the external gear in the tooth trace direction hasundergone shifting in the aforedescribed manner. In the drawing, a statein a cross-section that includes the major axis with the gears 2, 3 in ameshed state (state of maximum-depth meshing) is shown. In a centerportion in the tooth trace direction that includes the principalcross-section 34 c, the tooth profile outline in the tooth tracedirection of the external gear 34 is specified by the aforedescribedquartic curve C1, the tooth profile outline in a section from thiscenter portion to the external-teeth open end portion 34 a is defined bythe straight shifting line L1, and the tooth profile outline in asection from this center portion to the external-teeth inner end portion34 b is defined by the straight shifting line L2.

FIGS. 8-10 are descriptive diagrams showing, by rack approximation, thecondition of meshing of the external teeth 34 with respect to theinternal teeth 24 having tooth profiles established in theaforedescribed manner. FIG. 8 shows meshing of the external teeth 34with respect to the internal teeth 24 in the external-teeth open endportion 34 a of the external gear 34. FIG. 9 (a) shows analogous meshingin the principal cross-section 34 c of the external gear 34, and FIG. 9(b) is a partial enlarged view thereof. FIG. 10 shows analogous meshingin the external-teeth inner end portion 34 b of the external gear 34.

As will be understood from the drawings, while approximate, at locationsfrom the external-teeth open end portion 34 a to the external-teethinner end portion 34 b of the external gear 3, the tooth profiles makeeffective contact, centered on the principal cross-section 34 c.

As described above, in the present example, by making necessarycorrections to the tooth profile of the flexible external gear 3 of thestrain wave gearing 1, in an axis-perpendicular cross-section having adeflection coefficient of κ=1 (the principal cross-section 34 c), thelocation of meshing of the external gear 3 with respect to the internalgear 2 in the external gear 3 is moved away from the location of themajor axis La of the ellipsoidal rim neutral curve of the external gear3, and gradual meshing commences. In so doing, superimposition ofbending stress produced by flexion, and tensile stress caused by loadtorque, arising at major axis locations of the ellipsoidal rim neutralcurve of the external gear as encountered in the prior art, can beavoided.

In particular, the positions where the two stresses (bending stress andtensile stress) arise can be separated substantially and completely,whereby the transmission torque capacity of the strain wave gearing canbe improved, without the need to adopt negative deflection flexinghaving a deflection coefficient of κ<1 in a flat-type strain wavegearing, or to adopt negative deflection flexing having a deflectioncoefficient of κ<1 along the entire tooth profile in a cup-type ortop-hat-type strain wave gearing.

Further, according to the present invention, tooth shifting is adoptedfor the external gear other than principal cross-section thereof,whereby realizing continuous meshing between the external gear and theinternal gear along the tooth trace direction in a cup-type ortop-hat-type strain wave gearing. This can further increase thetransmission torque capacity of a strain wave gearing.

The invention claimed is:
 1. A strain wave gearing comprising: a rigidinternal gear, a flexible external gear arranged coaxially inside of therigid internal gear, and a wave generator fitted inside the flexibleexternal gear; wherein the external gear is flexed into an ellipticalshape by the wave generator, and external teeth of the ellipsoidallyflexed external gear mesh with internal teeth of the internal gear inregions avoiding sections at opposite ends of the ellipsoidally flexedexternal gear in a major axis direction thereof; the internal gear, andthe external gear, both are spur gears of module m; a number of teeth ofthe external gear is fewer by 2n than a number of teeth of the internalgear, where n is a positive integer; at a location along the major axison an ellipsoidal rim neutral curve of the external gear in anaxis-perpendicular cross-section at a predetermined location along atooth trace direction of the external gear, a radial flexing amount withrespect to a rim neutral circle prior to flexion is 2κmn, where κ is adeflection coefficient, and where an axis-perpendicular cross-sectionestablished at a predetermined location lying in the tooth tracedirection of the external gear is a principal cross-section, theprincipal cross-section being a non-deflection cross-section in whichthe deflection coefficient κ=1; a movement locus where the deflectioncoefficient κ=1 by the teeth of the external gear with respect to theinternal gear, and where meshing of the external gear with respect tothe internal gear in the principal cross-section comprises rack meshing;a tooth profile of an addendum of the internal gear is specified by thefollowing formula a,x _(Ca1)=0.25 mn(π+θ−sin θ)y _(Ca1)=0.5 mn(−1+cos θ)  (formula a)  where 0≤θ≤π; a tooth profile ofan addendum of the external gear is specified by the following formulab,x _(Fa1)=0.25 mn[π−θ+sine θ−ε{cos(θ/2)−sin(θ/2)}]y _(Fa1)=mn[0.5(1−cos θ)−(ε/4){sin(θ/2)−cos(θ/2)}]  (formula b)  where0≤ε≤0.1 and 0≤θ≤π; and the tooth profiles of dedenda of each of theinternal gear and the external gear are set to any shape that does notinterfere with the tooth profile of the addendum of the other gear. 2.The strain wave gearing according to claim 1, wherein a dedendum profileof the internal gear at a location of its maximum tooth thickness isgiven by the following formula c,x _(Ca2)=0.25 mn(π−θ+sin θ)y _(Ca2)=0.5 mn(1−cos θ)}  (Formula c)  where 0≤θ≤π; and a dedendumprofile of the external gear at a location of its maximum tooththickness is given by the following formula d,x _(Fa2)=0.25 mn[π−θ+sin θ−ε{cos(θ/2)−sin(θ/2)}]y _(Fa2)=mn[0.5(−1+cos θ)−(ε/4){sin(θ/2)−cos(θ/2)}]  (Formula d)  where0≤ε≤0.1 and 0≤θ≤π.
 3. The strain wave gearing according to claim 1,wherein tooth profiles of an addendum of axis-perpendicularcross-sections along the tooth trace direction of the internal gear aredefined by the above formula a; and tooth profiles of addendum ofaxis-perpendicular cross-sections in the tooth trace direction of theexternal gear are defined by the above formula b.
 4. The strain wavegearing according to claim 1, wherein the external gear has a flexiblecylindrical body part, and a diaphragm extending in a radial directionfrom a back end of the cylindrical body part, the external teeth beingformed in an outer peripheral section at a front open-end side of thecylindrical body part; a flexing amount of the external teeth changesrelative to a distance from the diaphragm from an end of the externalteeth adjacent the diaphragm towards an open end of the external teethat the front open-end side in the tooth trace direction; the principalcross-section is located at a center along the tooth-trace-directionbetween the external-teeth open end and the external-teeth inner end ofthe external teeth; the tooth profile of the external gear in theprincipal cross-section is defined by an addendum profile that isdefined by the above formula b; and the tooth profile inaxis-perpendicular cross-sections, other than the principalcross-section, along the tooth trace direction in the external gear areshifted profiles in which the tooth profile of the principalcross-section is subjected to shifting according to the flexing amountof each of the axis-perpendicular cross-sections, and wherein the toothprofiles of axis-perpendicular cross-sections of the tooth tracedirection, from the principal cross-section to the external-teeth openend of the external gear, are obtained by subjecting the tooth profileof the principal cross-section to shifting, in such a way that apexportions of the movement locus where the deflection coefficient κ>1described by the tooth profile in each of the axis-perpendicularcross-sections contact apex portions of the movement locus where thedeflection coefficient κ=1 in the principal cross-section; and the toothprofiles of axis-perpendicular cross-sections of the tooth tracedirection, from the principal cross-section to the external-teeth innerend of the external gear, are obtained by subjecting the tooth profileof the principal cross-section to shifting, in such a way that nadirportions of the movement locus where the deflection coefficient κ<1described by the tooth profiles in the axis-perpendicular cross-sectionscontact nadir portions of the movement locus where the deflectioncoefficient κ=1 in the principal cross-section.
 5. The strain wavegearing according to claim 4, wherein the tooth profiles ofaxis-perpendicular cross-sections of the tooth trace direction, from theprincipal cross-section to the external-teeth open end of the externalgear, are obtained by shifting the tooth profile of the principalcross-section, the amount of shifting being defined by the followingformula,h=κ cos θ_(κ)−cos θ₁,  where values of θ_(κ) and θ₁ are solutions of thefollowing simultaneous equations,(1−κ cos θ_(κ))/κ sin θ_(κ)−(1−cos θ₁)/sin θ₁=0θ_(κ)−κ sin θ_(κ)−θ₁+sin θ₁=0.
 6. The strain wave gearing according toclaim 5, wherein the tooth profiles of axis-perpendicular cross-sectionsalong the tooth trace direction, from the principal cross-section to theexternal-teeth inner end of the external gear, are obtained by shiftingthe tooth profile of the principal cross-section, the amount of shiftingbeing defined by the following formula,h=κ−1.